Dihedral angle of a regular simplex in $n$ dimensions

Question

For the regular simplex on $(n+1)$ points in $n$ dimensions, what is the dihedral angle i.e. the angle between two of the faces?

Answer 1

It's $\cos^{-1}(1/n)$. To compute it, for concreteness say the simplex is formed as the convex hull of $(n+1)$ unit vectors $\{u_i\}$, and the faces are $P_i$ with $P_i$ opposite to $u_i$. Let $\alpha$ be the angle between $u_1, u_2$. It's actually the case that $\pi - \alpha$ is the angle we're looking for: $u_1$ and $u_2$ are normal vectors to $P_1$ and $P_2$, and the angle between normal vectors is the same as between the planes (except $\alpha$ is the obtuse angle between the planes, hence the need to subtract it from $\pi$).

To compute $\alpha$, consider the triangle formed by $u_1$, $u_2$, and the origin. Let $d$ be the length of the edge between $u_1$ and $u_2$. By the Law of Cosines, $$d^2 = \|u_1\| + \|u_2\| - 2\|u_1\|\cdot\|u_2\| \cos\alpha = 2 - 2\cos \alpha $$ So it suffices to compute $d$. By an explicit computation for the standard simplex as done exactly in this answer, a simplex with circumradius 1 has edge length $d = \sqrt{\frac{2(n+1)}{n}}$. So $$ \frac{2(n+1)}{n} = 2 - 2\cos \alpha$$ $$ \alpha = \cos^{-1}(-1/n) = \pi - \cos^{-1}(1/n)$$ $$ \pi - \alpha = \cos^{-1}(1/n)$$

This answer is a simplification of "An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex" by Parks and Wills.